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Theorem sbcrext 2829
Description: Interchange class substitution and restricted existential quantifier. (Contributed by NM, 1-Mar-2008.) (Proof shortened by Mario Carneiro, 13-Oct-2016.)
Assertion
Ref Expression
sbcrext ((A 𝑉 yA) → ([A / x]y B φy B [A / x]φ))
Distinct variable groups:   x,y   x,B
Allowed substitution hints:   φ(x,y)   A(x,y)   B(y)   𝑉(x,y)

Proof of Theorem sbcrext
Dummy variable z is distinct from all other variables.
StepHypRef Expression
1 sbcco 2779 . 2 ([A / z][z / x]y B φ[A / x]y B φ)
2 simpl 102 . . 3 ((A 𝑉 yA) → A 𝑉)
3 sbsbc 2762 . . . . 5 ([z / x]y B φ[z / x]y B φ)
4 nfcv 2175 . . . . . . 7 xB
5 nfs1v 1812 . . . . . . 7 x[z / x]φ
64, 5nfrexxy 2355 . . . . . 6 xy B [z / x]φ
7 sbequ12 1651 . . . . . . 7 (x = z → (φ ↔ [z / x]φ))
87rexbidv 2321 . . . . . 6 (x = z → (y B φy B [z / x]φ))
96, 8sbie 1671 . . . . 5 ([z / x]y B φy B [z / x]φ)
103, 9bitr3i 175 . . . 4 ([z / x]y B φy B [z / x]φ)
11 nfnfc1 2178 . . . . . . 7 yyA
12 nfcvd 2176 . . . . . . . 8 (yAyz)
13 id 19 . . . . . . . 8 (yAyA)
1412, 13nfeqd 2189 . . . . . . 7 (yA → Ⅎy z = A)
1511, 14nfan1 1453 . . . . . 6 y(yA z = A)
16 dfsbcq2 2761 . . . . . . 7 (z = A → ([z / x]φ[A / x]φ))
1716adantl 262 . . . . . 6 ((yA z = A) → ([z / x]φ[A / x]φ))
1815, 17rexbid 2319 . . . . 5 ((yA z = A) → (y B [z / x]φy B [A / x]φ))
1918adantll 445 . . . 4 (((A 𝑉 yA) z = A) → (y B [z / x]φy B [A / x]φ))
2010, 19syl5bb 181 . . 3 (((A 𝑉 yA) z = A) → ([z / x]y B φy B [A / x]φ))
212, 20sbcied 2793 . 2 ((A 𝑉 yA) → ([A / z][z / x]y B φy B [A / x]φ))
221, 21syl5bbr 183 1 ((A 𝑉 yA) → ([A / x]y B φy B [A / x]φ))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wb 98   = wceq 1242   wcel 1390  [wsb 1642  wnfc 2162  wrex 2301  [wsbc 2758
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-rex 2306  df-v 2553  df-sbc 2759
This theorem is referenced by: (None)
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